To further quantify the complexity of dynamical systems of infinite entropy, a new invariant called topological mean dimension was introduced by Gromov [1]}. Metric mean dimension was introduced later by Lindenstrauss and Weiss [2]} as an upper bound for topological mean dimension. Recently, a double (minimax) variational principle is obtained for topological mean dimension by Lindenstrauss and Tsukamoto [3]}, and is extended to topological mean dimension with potential by Tsukamoto [4]}. Lindenstrauss and
Tsukamoto [5]} also established a variational principle formulating metric mean dimension as a supremum of certain rate distortion
functions over invariant measures of the system. Alternative formulations of metric mean dimension in terms of metric entropy of partitions, Brin-Katok local entropy, Katok's entropy and Shapira's entropy are developed by Velozo-Velozo [6]}, Gutman and Śpiewak [7]}, and Shi [8]}.
